Abstract

A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrödinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg–de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.

1. Introduction

Dark solitary waves (DSWs) are waves that are bi-asymptotic to a periodic state, with modulation in between. They are well known as exact solutions of the defocusing nonlinear Schrödinger (NLS) equation,
1.1for the complex-valued function A(x,t). An example of a DSW solution is
1.2where with ω>0. With ω fixed, the above-mentioned family of DSWs is a one-parameter family parametrized by k, bi-asymptotic to a spatially periodic state, with
The background spatially periodic state exists for . The DSW emerges when 3k2=ω and it exists in the narrower band .

The purpose of this paper is to develop a theory for this emergence of a DSW from a spatially periodic state. The main interest is in a theory for this phenomenon in Hamiltonian PDEs that are not, in general, integrable. However, an idea of the mechanism that leads to emergence can be clearly seen by considering the case of the defocusing NLS equation.

The first theory for the emergence of DSWs was proposed by Kivshar (1990). Kivshar's theory starts with the defocusing NLS equation and proposes a perturbation
where
1.3Expanding ϕ and u in a perturbation series in ε and imposing a solvability condition leads to a Korteweg–de Vries (KdV) equation for the leading-order term u1,
where the coefficients a0, a1 and a2 depend on (ω,k); that is, the emergence of DSWs is governed by a KdV equation. Kivshar et al. (1993) extended this theory to the non-integrable NLS equation where the cubic nonlinearity is replaced by a general nonlinearity f(|A|2)A. A validity proof of the reduction NLS → KdV for the general nonlinearity is proved by Chiron & Rousset (2010) (see also Béthuel et al.2009, 2010). Moreover, they extend the theory to the defocusing NLS equation in two-space dimensions; in that case, the reduction is .

One feature that is hidden in this theory is the bifurcation that the spatially periodic state must undergo in order to generate the KdV equation. The fact that ‘there must be a bifurcation’ is clear because the generic modulation of a spatially periodic state (or spatially periodic travelling wave) is governed by the Whitham modulation equations that are dispersionless. Indeed, Düll & Schneider (2009) prove that the dispersionless Whitham equations are the valid modulation equation for the spatially periodic states of the defocusing NLS equation.

In this paper, a bifurcation that gives rise to DSWs is identified. It is the saddle–centre transition of the Floquet multipliers associated with the spatially periodic states. When considered from the point of view of a (spatial) energy surface, the transition is associated with the coalescence between two spatially periodic states, of which one is spatially elliptic and the other spatially hyperbolic.

The theory will be developed for a general class of Hamiltonian PDEs. The principal requirement is that the steady part is a standard (in general, not integrable) Hamiltonian system on ,
1.4where J is the unit symplectic operator
1.5and S(u) is the (spatial) Hamiltonian function. The symbol S is used to distinguish it from a temporal Hamiltonian function. Phase space dimension 4 is the lowest dimension in which the phenomenon occurs. Generalization to a higher dimension is possible with appropriate hypotheses on the additional Floquet multipliers (for discussion, see §6). To make it a PDE, add in a time-derivative term. A large class of Hamiltonian PDEs arises by taking the time-derivative term to be a skew–symmetric matrix times the time derivative
1.6M is any constant matrix with MT=−M. The skew–symmetry of M assures that the PDE is conservative. Indeed, this system is an example of a multi-symplectic Hamiltonian PDE (Bridges 1997a,b; Bridges et al. 2010). Here, the only property of the time-derivative term that will be important is the skew–symmetry of M. Examples that can be expressed in the form (1.6) are the nonlinear beam equation, the NLS equation, the good Boussinesq equation and the coupled-mode equation. These examples are discussed in §6 and in appendix D.

The main result of the paper is that the emergence of DSWs is governed by a KdV equation of the following form:
1.7for the wavenumber perturbation q(X,T), where X and T are the KdV variables (1.3). This q-KdV equation is on a periodic background. Moreover, the coefficients in the KdV equation are determined by the geometry of the colliding periodic orbits. is the wave action, is the wave action flux and is the Krein signature of the elliptic periodic orbit in the collision. These geometrical properties are developed in §5.

In general Hamiltonian PDEs, the emerging waves may be of other related types (e.g. ‘grey solitary waves’), but, for the sake of simplicity, the term DSW will be used throughout, because the emergence has a universal form. DSWs can be stationary or moving at some constant speed, and the periodic state at infinity can be stationary or moving at some speed. For definiteness in this paper, the spatially periodic state is taken to be stationary, and the KdV equation is also constructed relative to an absolute (laboratory) frame of reference.

This theory is to be contrasted with the modulation of spatially periodic states of reaction–diffusion (RD) equations, which requires a time scaling of T=ε2t and leads to a Burgers equation for the perturbation wavenumber q (Doelman et al. 2009). However, a KdV equation can appear in the RD setting when the wavetrain undergoes modulation instability, and then the modulation equation changes to a KdV equation (van Harten 1995; Doelman et al. 2009). The appearance of the KdV equation in the setting of RD equations is more remarkable because the KdV equation is Hamiltonian and RD equations are, in general, dissipative. On the other hand, the absence of Hamiltonian structure and the conservation of wave action lead to a different form of the KdV equation in the RD setting.

The paper is organized as follows. First, the theory of Kivshar is reviewed in §2. Then, with the assumption that there is a branch of periodic states of (1.4), the properties of these states and the linearization about them is reviewed in §3.

The reduction theory from (1.6) to q-KdV (1.7) is carried out in §4. There are two steps to this theory: expansion in powers of ε, and expansion in polynomials of the coordinates of the linearized solution. A theory, particularly conservation of wave action, that leads to the geometrical characterization of the coefficients of the KdV equation is developed in §5. Examples are discussed in §6 and appendix D. The appendices record the details of some of the technical calculations, the definition of the Krein signature, the geometrical formulation of the conservation of wave action and the transformation of the coupled-mode equation to the form (1.6).

2. Kivshar's theory for the reduction of the nonlinear Schrödinger equation to the Korteweg–de Vries equation

In this section, the theory of Kivshar (1990) is reviewed, with an emphasis on the case where the basic state is spatially periodic. Take, as the starting point, the following form of the defocusing NLS equation:
2.1The basic class of steady periodic solutions is
2.2Look for solutions nearby the periodic state of the form
Substitute into (1.1), separate real and imaginary parts, and introduce the KdV scaling (1.3)
2.3and
2.4Expand ϕ and u in a power series in εSubstitution then gives the following equations to leading order. For the u-equation (2.3),
2.5and for the ϕ-equation (2.4),
2.6Differentiate and combine the ε3 term in (2.5) and the ε2 term in (2.6),
For arbitrary u1, this is satisfied if and only if
2.7The significance of this condition will be discussed later in the text. When this condition is satisfied, u1 and ϕ1 are related by
2.8

Now, look at the next terms in (2.3) and (2.4), and replace ∂ϕ1/∂X using (2.8),
and
With the condition (2.7), this pair of equations is solvable if and only if
giving a KdV equation

(a) The Madelung transformation

Another approach to the reduction from the NLS equation to the KdV equation is to use the Madelung transformation. Introduce the transformation A=ρ(x,t)eiϕ in (1.1) and let
Then, h(x,t) and u(x,t) satisfy
which are the shallow water equations with a complicated dispersion term. Near a constant state (h0,u0), these equations can be reduced to a KdV equation when by unidirectionalization. This approach is used in the rigorous reduction theory of Chiron & Rousset (2010) and Béthuel et al. (2009, 2010).

(b) The saddle–centre transition

In the derivation of the KdV equation from the NLS equation, a necessary condition was (2.7). Let us look more closely at the significance of this condition. The amplitude of the branch of periodic solutions satisfies (2.2), and this curve is shown in figure 1. The points are points where the spatial Floquet multipliers go through a hyperbolic–elliptic (saddle–centre) transition. For , the branch of periodic solutions is hyperbolic. These features can be verified by direct calculation.

A plot of versus k for the branch of periodic solutions. The change from the solid line to the dashed line occurs at .

However, the presentation in figure 1 misses the key features of the bifurcation. Consider the ‘spatial’ energy for the defocusing NLS equation,
(Sx=0 for the steady part of (2.1)), and evaluate on the branch of periodic solutions (2.2)
2.9The spatial energy is plotted in figure 2. This diagram shows more clearly the structure of the bifurcation. The two maximum points occur at with the critical value . For values of S just below the critical value, there are two periodic solutions with k>0—one is (spatially) elliptic and one (spatially) hyperbolic—and, as from below, they coalesce, giving birth to the DSW.

On the other hand, it is the wave action flux that will be important in the development of the theory. The wave action flux is defined in §5 and in appendix B, but, for the purposes of this section, it takes the form when evaluated on the branch of periodic states. It is plotted in figure 3 as a function of the wavenumber. In this figure, it can be interpreted that, as approaches its maximum for k>0, an elliptic periodic state collides with a hyperbolic periodic state. These features are general properties of an elliptic–hyperbolic transition of Floquet multipliers along a branch of periodic solutions for a Hamiltonian system on , and general aspects of this bifurcation are developed in §3.

3. Periodic states and the saddle–centre transition

The starting point is the system (1.4). Suppose that there exists a one-parameter family of periodic solutions of (1.4),
3.1where k>0 is the wavenumber of the periodic state, θ0 is an arbitrary phase shift and is a 2π-periodic function of θ. Substituting (3.1) in (1.4) shows that the periodic orbit satisfies
3.2Define the ‘action’ functional by
3.3Although this functional is the action for the Hamiltonian ODE (1.4), it will be shown later that it is the wave action flux associated with the time-dependent Hamiltonian PDE (1.6).

With the definition (3.3), the governing equation (3.2) can be interpreted as the Euler–Lagrange equation for the constrained variational principle: find critical points of the Hamiltonian function restricted to level sets of the action, with k appearing as the Lagrange multiplier. This variational principle is difficult to work with because all the critical points have infinite Morse index. On the other hand, the structure from the variational principle will be useful. From the theory of Lagrange multipliers, it follows that a critical point is non-degenerate if (equivalently ), where
3.4At points where changes sign, a pair of Floquet multipliers coalesces at +1. At points where the energy is stationary, the action is also stationary,
Hence, and k≠0 imply that . Here, the angular brackets denote the inner product
3.5for any -valued 2π-periodic functions v(θ) and w(θ). The link between Floquet multipliers and curves is shown in figure 4.

Schematic of the elliptic–hyperbolic transition of Floquet multipliers.

There are four cases, and they can each be represented in a diagram as shown in figure 5. Because at the transition, the curves are locally parabolic. Let κ represent the curvature of at the transition (a precise definition of κ is given in §4). In each case, an elliptic branch meets a hyperbolic branch, and the elliptic branch can have a positive or negative Krein signature. The four cases are illustrated in figure 4. An explicit calculation of the Krein signature, justifying the choices in the figure, is given in appendix C.

Four cases in the elliptic–hyperbolic transition: h denotes the hyperbolic branch of the periodic orbits, and e± denotes the elliptic branch with Krein signature ±1.

(a) Linearization at the saddle–centre transition

Now suppose there is a saddle–centre transition of eigenvalues, and look at the linearization of the spatial Hamiltonian system (1.4) about the family of periodic solutions (3.1).

At the transition, the algebraic multiplicity of the Floquet multiplier at +1 is (at least) 4 and the geometrical multiplicity is 1, but phase space dimension 4 ensures that the algebraic multiplicity equals 4. The Jordan chain theory for this case is given in Bridges & Donaldson (2005).

First, the linearization about a periodic orbit of an autonomous Hamiltonian system always has a Floquet multiplier at +1 of geometrical multiplicity 1 and algebraic multiplicity 2. This property follows from differentiation of (3.2) with respect to θ and k,
3.6with , and
3.7The consequence of the degeneracy is that the algebraic multiplicity of the Floquet multiplier +1 is increased to at least 4. In this case, there exist functions

The solutions , j=1,2,3,4 are not unique, and they do not form a symplectic basis with respect to the operator J. Introduce a normalized basis
where
3.8and
3.9This set of vectors still forms a Jordan chain:
3.10In the transformed basis
3.11The transformation matrix
3.12is symplectic with respect to J: ΣTJΣ=J. Symplecticity of this transformation is not essential, but it results in the reduced system, leading to q-KdV, having the same spatial symplectic structure as the original system, and it is used in the definition of the Krein signature. The purpose of the sign s is to ensure that Σ in (3.12) is symplectic. The sign s is also a symplectic invariant. (Because s is defined via the symplectic form in (3.11), any symplectic change of coordinates leaves it invariant.)

By taking coordinates (ϕ,q,I,p), the solution of the linear problem in the neighbourhood of the elliptic–hyperbolic transition is
3.13In the later calculations, equations for the second derivatives of will be needed. They are obtained by differentiating the linear equations (3.6),
3.14The formal definition of the third derivative at a point is
3.15where ϵ=(ϵ1,ϵ2,ϵ3), and the basepoint is omitted because the context is clear. The right-hand side is invariant under permutation of the three vectors.

4. Scaling, expansion and reduction to the Korteweg–de Vries equation

Start with the time-dependent Hamiltonian PDE (1.6) and perturb about the spatially periodic state near a saddle–centre transition,
and substitute into (1.6)
4.1Drop the tilde on the perturbation, as the context will be clear, and introduce the KdV scaling (1.3),
4.2where L is the linear operator from §3, and the nonlinear term is
4.3Henceforth, the derivatives of S will be written without the base point identified as in (3.15).

The perturbed solution u(θ,X,T,ε)—now considered a function of the slow time and space variables instead of x,t—is expanded in a power series in ε. A key step is to scale the Jordan chain coordinates in (3.13) as
4.4Hence, the scaled solution for the linear terms is
Now, expand the solution u as a polynomial in the coordinates (ϕ,q,I,p) and with the scaling (4.4)
4.5with
In this expansion, the functions ψj, j=1,…,6, χj, j=1,…,4 and γj, j=1,2,3 are unknown functions at this point and are used to eliminate terms in the expansion.

Substitute the expansion (4.5) into (4.2) and equate terms proportional to like powers of ε to zero,
4.6

where represents terms that are of cubic and higher order in (ϕ,q,I,p). In the third equation, the term is added to the left-hand side with a parameter κ (to be determined), and the same term is added to on the right-hand side. This addition is the natural way to ensure that can be set to zero. The terms on the right-hand side are
4.7The strategy is to show that it is possible to set these six equations to zero, by solving for ψj, j=1,…,6. Because L is symmetric and has a non-trivial kernel, setting each term to zero involves satisfying a solvability condition. Remarkably, all six equations are solvable if and only if
4.8This expression for κ is important because it will appear later as the coefficient of the nonlinear term in the KdV equation. A proof of this result is in appendix A.

Set each of the terms in (4.7) to zero in the four equations (4.6), and take the inner product of each equation with ξ4,…,ξ1 in turn, and use (3.11),
Define
4.9The skew–symmetry MT=−M then gives m21=−m12. Now, substitute for (pX+I) from the third equation into the fourth
4.10The next step is to show that the term proportional to ϕϕT can be transformed away. This can be carried out by introducing a transformation . Substitution into (4.10) leads to the following equation for b:
After some simplification, the equation for b reduces to
Hence, if m12≠0, this equation uniquely defines b.

After transformation, neglect of the cubic terms, neglect of terms of order ε6 and higher, and re-ordering of the equations to emphasize the symplectic formulation of the steady part, the reduced equation is
4.11To see that (4.11) is the KdV equation, differentiate the second equation with respect to X, and eliminate ϕ, I and p reducing it to
4.12It remains to show that the coefficients in this KdV equation have a geometrical interpretation, and this is done in §5.

5. Geometry of the coefficients of the Korteweg–de Vries equation

In this section, the coefficients of the KdV equation (4.12) are given a geometrical interpretation. Start with the coefficient of the time-derivative term. From the definition (4.9)
Using the definition of in appendix B,
after integration by parts and using periodicity. But the latter term is a−2m12, and so
5.1

To prove that the coefficient κ in (4.11) is proportional to the second derivative of the wave action flux, start with the definition of wave action flux in appendix B and differentiate,
5.2Taking the second derivative
5.3The key is to note that and , and then use the equations for the second derivatives (3.14). First define
5.4and note that it is invariant under permutation of its three indices (see (3.15)). Calculating,
and so
Comparison with (4.8) shows that . Substitute into (4.12),
Now use the fact that
and then a scaling of the form eliminates a, and a scaling of time eliminates the factor 2, leaving
Now a uniform scaling of space and time and setting
then leads to the normal form of the q-KdV equation in (1.7). The connection between the Krein signature and the symplectic sign s is given in appendix C.

6. Concluding remarks

The assumption of phase space dimension 4 in (1.4) and (1.6) can be relaxed, depending on the position of the other Floquet multipliers. If all the other Floquet multipliers are hyperbolic (even if there are an infinite number), then the theory goes through with a centre-manifold-type reduction of the hyperbolic Floquet multipliers. If at least one pair of other Floquet multipliers is elliptic, then the theory changes, and it is likely that the KdV equation (1.7) will be accompanied by another modulation equation.

Nowhere in the formal construction and reduction to KdV is well-posedness of (1.6) used. Indeed, (1.6) can even be elliptic in time. However, for any validity result involving evolution in time, well-posedness of (1.6) will be necessary.

The basic spatially periodic state was assumed stationary relative to an absolute frame of reference. Suppose that it is stationary relative to a frame of reference moving at speed c. Then, (1.6) is replaced by
6.1and then the theory goes through as before with the steady system (1.4) being replaced by the c-dependent steady system in (6.1). The moving frame just brings in an additional parameter.

A key structural requirement for the theory is that the PDEs can be formulated as in (1.6). Some examples are (i) the defocusing NLS equation (1.1), (ii) the good Boussinesq equation
and (iii) the coupled-mode equation (see appendix D). Details of specific examples will be considered elsewhere.

There are potential applications in the theory of water waves. Unsteady DSWs appear in the theory of water waves when the depth is sufficiently shallow, because the NLS model for water waves is defocusing in that regime (see Hasimoto & Ono 1972; Infeld & Rowlands 2000).

However, the theory in this paper suggests that DSWs will emerge at finite amplitude if a saddle–centre transition of eigenvalues occurs in the linearization about travelling waves. Indeed, this is the case. Vandenbroeck (1983) computes spatial Floquet multipliers along a branch of Stokes waves and notes the occurrence of a saddle–centre transition. At low amplitudes, coupled to a mean flow, Bridges & Donaldson (2006) analytically find saddle–centre transitions. In Bridges & Donaldson (2006), the saddle–centre transition is used as a basis for showing that steady DSWs can emerge. The theory in this paper shows that this result can be improved by showing that the unsteady DSWs can also emerge.

Appendix A. Proof that solvability leads to κ

Because L is formally symmetric, with ξ1 in the kernel, it follows that
(A1)Applying this to the six equations in (4.7) results in the following two conditions:
(A2)and
(A3)Equation (A3) is satisfied identically. To verify (A2), consider solvability of the fifth equation in (4.7), and use the symmetry of L, (3.10) and the second derivatives (3.14) repeatedly,
Noting that the last two terms cancel out, divide by s and use (3.11), then solvability requires
after repeated use of (3.10) and (3.14). This confirms the expression for κ in (A2).

Appendix B. Conservation of wave action

Systems of the type (1.6) have a geometrical formulation of conservation of wave action (see Bridges 1997a,b). Consider an ensemble of solutions u(x,t,θ) of (1.6), which is 2π-periodic in the ensemble parameter θ. Define wave action and wave action flux as
(B1)Conservation of wave action, At+Bx=0, follows from

Now, evaluate the wave action and wave action flux on solutions with θ=kx+θ0, and normalize the integrals by 2π,
(B2)It is the derivatives with respect to k of these two functions that appear as coefficients in the KdV equation.

Appendix C. Krein signature and the symplectic sign s

Here, the connection between the symplectic sign s and the Krein signature of the elliptic periodic orbit in the collision is established, and the choice of signs e± in figure 4 is justified.

The curve of periodic solutions appears in the reduced system as relative equilibrium solutions of (4.11); that is,
with q0 and I0 constants. The (q0,I0) curve is a parabola, and there are two cases depending on the sign of κ. However, taking into account that there is a change from hyperbolic to elliptic at the maximum (or minimum) of the parabola, there are four cases and they are shown in figure 5. It remains to establish which branches are elliptic and hyperbolic and the Krein signature of the elliptic branch.

To determine ellipticity or hyperbolicity of the branch of periodic solutions, only the steady system is required. Linearize the reduced system (4.11) dropping the time derivatives
(C1)Taking solutions proportional to eμx results in the characteristic equation
Hence, the non-trivial exponents are
and they are hyperbolic if sκq0>0 and elliptic if sκq0<0. This confirms the labelling of hyperbolic and elliptic in figure 5. For figure 5, q0 should be interpreted as k−k0, with , and I0 should be interpreted as .

Now, consider the elliptic case where −sκq0>0, and define . The complex eigenvector ζ corresponding to μ=iσ is
(C2)Define the Krein signature () associated with a purely imaginary eigenvalue, with eigenvector ζ, in the linearization about an equilibrium, to be
(C3)The eigenvector ζ in (C 2) has been scaled so that the sign . Computing the Krein signature for the eigenvalue iσ and eigenvector ζ gives
and so the Krein sign equals −s. The resulting Krein signs of each branch are then labelled in figure 5 as e±; that is, a branch with e+ has and a branch with e− has .

Appendix D. Multi-symplectic formulation of the coupled-mode equation

Consider M in (1.6) to be of the following form:
In this case, the operators M and J have the interesting properties
These properties ensure that the operator M∂t+J∂x is a d'Alembertian operator (symbol of a wave equation).

Taking S to be arbitrary, the system (1.6) has the form
with coordinates u=(u1,u2,v1,v2). A special case of interest is the coupled-mode equation (Grimshaw 2000; Grimshaw & Christodoulides 2001; Grimshaw & Skyrnnikov 2002; Derks & Gottwald 2005). To reduce to the coupled-mode equation, take the Hamiltonian function to be
Then, with
the earlier mentioned system can be written in complex form
which is the familiar form of the coupled-mode equation. When a1=a4=0, and parameters are scaled so that α=a2=1, the equation reduces to the integrable massive Thirring model (Derks & Gottwald 2005).

1997b A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R. Soc. Lond. A453, 1365–1395.doi:10.1098/rspa.1997.0075 (doi:10.1098/rspa.1997.0075)